MATH 220
NAME
EXAM 2
STUDENT NUMBER
SPRING 2011
INSTRUCTOR
VERSION A
SECTION NUMBER
On your scantron, write and bubble your PSU ID, Section Number, and Test Version. Failure
to correctly code these items may result in a loss of 5 points on the exam.
On your scantron, bubble letters corresponding to your answers on indicated questions. It
is a good idea for future review to circle your answers in the test booklet.
Check that your exam contains 30 multiple-choice questions, numbered sequentially.
Answer Questions 1–30 on your scantron.
Each question is worth 5 points.
THE USE OF A CALCULATOR, CELL PHONE, OR ANY
OTHER ELECTRONIC DEVICE IS NOT PERMITTED IN THIS
EXAMINATION.
THE USE OF NOTES OF ANY KIND IS NOT PERMITTED
DURING THIS EXAMINATION.
MATH 220
EXAM 2, VERSION A
PAGE 2
1. Determine h and k such that the solution set of the following system is empty.
x1 + 3x2 = k
4x1 + hx2 = 8
⋆ a) h = 12, k 6= 2
b) h = 12, k = 2
c) h 6= 12, k 6= 2
d) h 6= 12, k = 2






1
−3
h
2. Let v1 =  0 , v2 =  1 , and y =  −5 . For what value(s) of h is y in
−2
8
−3
Span{v1 , v2 }?
a) h =
3
2
b) h =
1
4
⋆ c) h = −
7
2
d) All real numbers
5 4
7 3
3. Suppose AB =
and B =
. Find A.
−2 3
2 1
1 −3
a) A =
−2
7
11 −17
b) A =
−27
41
13 43
c) A =
4 18
−3 13
⋆ d) A =
−8 27
MATH 220

EXAM 2, VERSION A


PAGE 3

a b c
a
b
c
4. If det d e f  = 8, what is det 3g + a 3h + b 3i + c ?
g h i
d
e
f
a) 8
⋆ b) −24
c) 24
d) −8
5. Find the determinant of






9
9
4
9
6
1
0
0
0
0
9
9
0
3
0
9
9
5
9
7
9
2
0
0
0



.


a) 0
b) −2
⋆ c) −12
d) 12
6. Let A be an invertible n × n matrix. Which of the following statements is not necessarily
true?
a) The equation Ax = b has at least one solution for each b in Rn .
b) AT is invertible.
c) Col A = Rn .
⋆ d) The number 0 is an eigenvalue of A.
MATH 220
EXAM 2, VERSION A
7. Let A =
⋆ a)
7
0
b)
6
0
c)
1
0
PAGE 4
6 1
. Which of the following matrices is similar to A?
3 4
0
3
0
4
0
3
d) None of the above.

−1
8. Which of the following is a unit vector in the same direction as v =  2 ?
1
 √ 
− 6
√


a)
6/3
√
6

 √
−√6/6
⋆ b)  √6/3 
6/6


−1/6
c)  1/3 
1/6

d) v is already a unit vector.
MATH 220
EXAM 2, VERSION A
PAGE 5
9. Which of the following statements is always true?
a) An n × n symmetric matrix has n distinct real eigenvalues.
b) An orthogonal matrix is orthogonally diagonalizable.
c) If P is an n × n matrix with orthogonal columns, then P T = P −1.
⋆ d) Every symmetric matrix is orthogonally diagonalizable.
10. Find the matrix of the quadratic form 8x21 + 7x22 − 3x23 − 6x1 x2 + 4x1 x3 − 2x2 x3 .


8 −3
2
7 −1 
⋆ a)  −3
2 −1 −3


8 −6
4
7 −2 
b)  0
0
0 −3


−3
4
7/2
2 −3/2 
c)  4
7/2 −3/2
−1


8
2 −3
7 −1 
d)  2
−3 −1 −3
11. Find the equation y = β0 + β1 x of the least-squares line that best fits the data points
(0, 1), (1, 1), (2, 2), (3, 2).
⋆ a) y = .9 + .4x
b) y = 4.5 + 9.5x
c) y = .4 + .9x
d) y = 9.5 − 4.5x
MATH 220
EXAM 2, VERSION A
PAGE 6
12. Find the characteristic polynomial of the following matrix:


−2 0
3
A =  1 5 −1  .
2 0
4
a) (−2 − λ)(5 − λ)(4 − λ)
⋆ b) (5 − λ)(λ2 − 2λ − 14)
c) λ3 + 5λ2 + 6λ − 14
d) 5(λ2 − 2λ − 14)

    
2
0
h
13. For what value of h are the vectors  2 ,  4 ,  5  linearly independent?
−1
0
6
a) h 6= 12
⋆ b) h 6= −12
c) h = −12
d) h = −10
2
2
14. Let T : R → R be a linear transformation such that T
1
2
−1
?
. What is T
2
5
8
a)
10
−8
b)
10
1
⋆ c)
−1
2
d)
−4
1
0
=
3
7
and T
0
1
=
MATH 220

EXAM 2, VERSION A

1
0 −1
0 , find the second column of A−1 .
15. If A =  3 −1
0 −1
1


1

1 
a)
−1
 
1
1
⋆ b)  1 
2 1


−1
c)  1 
−1


−1
1
d)  1 
2 −1
16. Find the coordinate
to
{v1 , v2 }

 vectorof x relative

 the basis

−3
7
−5
where x =  0 , v1 =  4 , v2 =  −2 .
5
−3
4
1
⋆ a)
2
−2
b)
1
−3
c)
2
−3
d)
−2
PAGE 7
MATH 220
EXAM 2, VERSION A
PAGE 8
17. What is the rank of a 3 × 6 matrix whose null space is 4-dimensional?
a) 1
⋆ b) 2
c) 3
d) 4

4 0 1
18. Find the eigenvalues of  −2 1 0 .
−2 0 1

a) 1, 1, 2
b) 1, 2, 2
c) 1, −1, 2
⋆ d) 1, 2, 3
19. Given the diagonalized matrix A =
compute A10 .
1
a)
2
310 − 1 310 + 1
310 + 1 310 − 1
1
b)
2
310 − 1 310 − 1
310 − 1 310 + 1
1
⋆ c)
2
310 + 1 310 − 1
310 − 1 310 + 1
1
d)
2
310 + 1 310 + 1
310 + 1 310 − 1
1 2
2 1
=
√ √ √
√
−1 0
−1/√2 1/√2
−1/√2 1/√2
,
0 3
1/ 2 1/ 2
1/ 2 1/ 2
MATH 220
EXAM 2, VERSION A



PAGE 9

3
2
20. What is the distance between u =  2  and v =  −1 ?
1
2
a) 3
√
b) 10
√
c) 12
√
⋆ d) 11
21. Find the orthogonal
, v2 }
1

 projection
 of x onto Span{v

−1
1
−1
where x =  4 , v1 =  1 , and v2 =  1 .
3
0
0


−1
⋆ a)  4 
0
 
3

1 
b)
0


0
c)  −1 
2


−2
d)  2 
1
MATH 220
EXAM 2, VERSION A
PAGE 10
22. If A is an n×n square matrix, and I denotes the n×n identity matrix, which of the following
statements is not necessarily true?
⋆ a) If the columns of A form an orthogonal set, then A is an orthogonal matrix.
b) If AAT = I, then A is an orthogonal matrix.
c) If A is an orthogonal matrix, its determinant must be ±1.
d) If A is an orthogonal matrix, the linear transformation x 7→ Ax preserves lengths and
orthogonality.
23. Which of the following statements is not true?
a) An n × n matrix that is orthogonally diagonalizable must be symmetric.
b) For any matrix A, AT A is symmetric.
c) If A is symmetric, any two eigenvectors of A corresponding to different eigenvalues
must be orthogonal.
⋆ d) If A and B are symmetric matrices, AB must also be symmetric.


 
1 3
2
24. Find the least-squares solution for the system Ax = b where A =  2 3  and b =  3 .
3 3
3
1/2
⋆ a)
5/9
1/2
b)
10/9
1/2
c)
7/9
1/2
d)
1/9
MATH 220
EXAM 2, VERSION A





PAGE 11



1
1
1
x
25. Let v1 =  1 , v2 =  −1 , v3 =  1 , and v =  y . Which of the following
1
0
−2
z
correctly express v as a linear combination of v1 , v2 and v3 ?
a) v =
x−y
x+y−z
x+y+z
v1 +
v2 +
v3
2
3
5
b) v =
x+y+z
x + y − 2z
x−y
v1 +
v2 +
v3
7
4
2
⋆ c) v =
x+y+z
x−y
x + y − 2z
v1 +
v2 +
v3
3
2
6
d) v =
x−y−z
x−y
x − y + 2z
v1 +
v2 +
v3
3
7
6


26. Let W = Span{v1 , v2 } where v1 = 

belong to the orthogonal complement



0
0


 8 
 and  2 
⋆ a) 
 0 
 0 
17
9




2
1


 12 
 and  17 
b) 
 0 
 0 
−1
0




2
2


 4 
 and  2 
c) 
 2 
 11 
−8
0




12
1


 0 
 and  0 
d) 
 −6 
 12 
0
0



2
1
 0
0 
 and v2 = 
 6
2 
0
0
of W ?


. Which of the following vectors

MATH 220
EXAM 2, VERSION A
PAGE 12


4 5 −11 2 5
27. Find a basis of the column space ColA for the matrix A =  4 5 −3 11 7 .
4 6 −4 3 1
a) The first and second columns
b) The first and last columns
⋆ c) The first, second and third columns
d) The fourth and fifth columns

3
28. Let v =  −3 . Which of the following vectors is in the same direction as v?
1


51
⋆ a)  −51 
17


−7
b)  −7 
3


−63
c)  63 
−21


57
d)  57 
−17

MATH 220
EXAM 2, VERSION A

PAGE 13

 
3
1
29. Let y = −1, v = 0. What is the distance from y to the line through 0 and v?
1
1
a) 1
√
b) 2
√
⋆ c) 3
d) 2
 
 
0
5
30. The vectors 4 and  6  form a basis for a subspace W . Use the Gram-Schmidt process
2
−7
to produce an orthogonal basis for W .
a)
⋆ b)
c)
d)
   
0
0
4 , 2
2
1
   
0
5
4 ,  4 
2
−8
   
0
5
3 ,  6 
1
−7
   
0
5
4 ,  6 
2
−7